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Munich Personal RePEc Archive
Spatial Period-Doubling Agglomeration
of a Core-Periphery Model with a
System of Cities
Ikeda, Kiyohiro and Akamatsu, Takashi and Kono, Tatsuhito
Department of Civil and Environmental Engineering, Tohoku
University
18 February 2009
Online at https://mpra.ub.uni-muenchen.de/25636/
MPRA Paper No. 25636, posted 05 Oct 2010 00:11 UTC
Spatial Period-Doubling Agglomeration of a
Core–Periphery Model with a System of Cities∗
Kiyohiro Ikeda†,Takashi Akamatsu and Tatsuhito Kono‡
February 18, 2009
Abstract
The orientation and progress of spatial agglomeration for Krug-man’s core–periphery model are investigated in this paper. Possibleagglomeration patterns for a system of cities spread uniformly on acircle are set forth theoretically. For example, a possible and mostlikely course predicted for eight cities is a gradual and successive one—concentration into four cities and then into two cities en route to asingle city. The existence of this course is ensured by numerical sim-ulation for the model. Such gradual and successive agglomeration,which is called spatial-period doubling, presents a sharp contrast withthe agglomeration of two cities, for which spontaneous concentrationto a single city is observed in models of various kinds. It exercises cau-tion about the adequacy of the two cities as a platform of the spatialagglomerations and demonstrates the need of the study on a systemof cities.
Keywords: Agglomeration of population, Bifurcation, Core–peripherymodel, Group theory, Spatial period doubling
JEL classification: F12; O18; R12
∗This research was supported by JSPS grants 19360227 and 21360240.†Address for correspondence: Kiyohiro Ikeda, Department of Civil and Envi-
ronmental Engineering, Tohoku University, Aoba 6-6-06, Sendai 980-8579, Japan;[email protected].
‡Graduate School of Information Sciences, Tohoku University Aoba 6-6-06, Sendai 980-8579, Japan
1
1 Introduction
Emergence of the spatial economic agglomeration attributable to market in-
teractions has attracted much attention of spatial economists and geogra-
phers. Among the many descriptions available in the literature, the core–
periphery model of Krugman (1991) [18] is touted as the first and the most
successful attempt to clarify the microeconomic underpinning of the spatial
economic agglomeration in a full-fledged general equilibrium approach1. The
core–periphery model introduced the Dixit–Stiglitz (1977) [6] model of mo-
nopolistic competition into spatial economics and provided a new framework
to explain interactions that occur among increasing returns at the level of
firms, transportation costs, and factor mobility. Such a framework paved
the way for development of the New Economic Geography2 as a mainstream
field of economics. Furthermore, in recent years, the framework has been
applied to various policy issues in areas such as trade policy, taxation, and
macroeconomic growth analysis (Baldwin et al., 2003 [1]).
Yet most reports of the literature in New Economic Geography have re-
mained confined to two-city models in which spatial economic concentration
to a single city is triggered by bifurcation3. The two-city model is the most
pertinent starting point by virtue of its analytical tractability, but it has a
limited capability to express spatial effects. In reality, economic agglomera-
tions can take place in more than two locations, as evidenced by results of
several empirical studies. For example, Behrens and Thisse (2007) [2] stated
that “Among a system of cities, indirect spatial effects emerge and compli-
cate the analysis. Dealing with these spatial indeterminacies constitutes a
1This is based on an appraisal by Fujita and Thisse (2009) [12] in honor of Krugman’s2008 Nobel Memorial Prize in Economic Sciences.
2Comprehensive reviews of the NEG models are available in a survey by Ottaviano andPuga (1998) [24], in a review by Fujita and Thisse (2009) [12], and in several books asfollows: Fujita et al. (1999) [10], Brakman et al., 2001 [3], Fujita and Thisse (2002) [11],Baldwin et al. (2003) [1], Henderson and Thisse (2004) [15], Combes et al. (2008) [4], andGlaeser (2008) [13].
3The two identical symmetric cities are in a stable state with high transport costs.When the costs are reduced to a certain level, tomahawk bifurcation triggers a spontaneousconcentration to a single city by breaking the symmetry (e.g., Krugman, 1991 [18], Fujitaet al., 1999 [10], Forslid and Ottaviano, 2003 [9]).
2
main theoretical and empirical challenge NEG and regional economics must
surely confront in the future4.” We must analyze a system of cities thoroughly
in careful comparison with the two-city model to answer the question, “To
what degree can we extrapolate the predictions and implications derived from
two-city analysis to a system of cities?”
Several reports in the literature have described attempts to transcend
the two-city special case with a local analysis (linearized eigenproblem) of
the racetrack economy5. Krugman (1993 [19], 1996 [20]) identified the emer-
gence of several spatial frequencies. Yet, currently, it is difficult analytically
to extract agglomeration properties from the nonlinear equations of core–
periphery models with an arbitrary discrete number of cities.
Numerical simulations might be effective for identifying agglomeration
patterns for a system of cities. A numerical simulation on 12 symmetric cities
of equal size is conducted to observe that the symmetric equilibrium often
becomes unstable (cf., Krugman, 1991 [18]). Fujita et al. (1999) [10] obtained
post-bifurcation equilibria for three cities. Nevertheless, it seems premature
to infer a global view of agglomeration based on currently available numerical
information. A naive numerical simulation for an increased number of cities
must address a rapidly increasing numerical information and might therefore
not be very promising.
The objective of this paper is to investigate the orientation and progress of
agglomerations of a system of cities and, in turn, to test the adequacy of the
two-city model as a spatial platform. Possible agglomeration patterns and
courses of the pattern change of a racetrack economy for the multi-regional
core–periphery model6 are obtained using group-theoretic bifurcation the-
4The difficulty encountered in solving the dimensionality problem is reminiscent of then-body problem in mechanics, which is solved for n = 2 but not for an arbitrary numberof bodies.
5The racetrack economy uses a system of identical cities that spread uniformly aroundthe circumference of a circle. See, e.g., Krugman (1993) [19], 1996 [20], Fujita et al. (1999)[10], Picard and Tabuchi (2009) [26].
6The core–periphery model with n cities is presented in §2 as a recapitulation and areorganization of Krugman (1991) [18] and Fujita et al. (1999, Chapter 5) [10]. Thismodel uses a spatial version of the Dixit–Stiglitz model, considers an economy with twosectors—agriculture and manufacturing— and assumes an upper-tier utility function ofthe Cobb–Douglas type with CES sub-preferences over manufacturing varieties.
3
Figure 1: Spatial period-doubling cascade for the eight cities (area of ⃝denotes the size of the associated city: the arrow denotes the occurrence ofa bifurcation)
ory7. A possible and the most likely course of agglomeration predicted is
spatial period-doubling cascade (cf. Proposition 5 in §5.4). An example of
this cascade is shown for eight cities in Fig. 1, in which the area of the circle
represents the population of the associated city and the arrow indicates the
occurrence of a bifurcation8. A system of 23 = 8 identical cities (for some
positive integer k) concentrates into 22 = 4 identical larger cities, en route to
the concentration to the single megalopolis. Consequently, the concentration
progresses successively in association with the doubling of the spatial period.
The validity of the theoretical prediction is assessed using numerical sim-
ulation. Basic equations of the core-periphery model are rewritten to be
compatible with computational bifurcation theory9 (cf. §3). A combination10
of the group-theoretic bifurcation theory and the computational bifurcation
theory is vital in the numerical simulation of the agglomerations of a race-
track economy with many cities.
Although Krugman’s (1993 [19], 1996 [20]) analysis of the racetrack econ-
omy gives the orientation of the breaking of uniformity, we study the progress
of agglomerations thereafter, as well as the orientation. The possible equi-
librium and associated agglomeration patterns of 4, 6, 8, and 16 cities are
studied theoretically (cf. §5) and are obtained numerically (cf. §6) in an ex-
haustive manner, while only a few were found and studied in the previous
numerical simulations. Results show that the racetrack economy among a
7The major framework of this theory has already been developed in physical fields (see,e.g., Golubitsky et al., 1988 [14]; Ikeda and Murota, 2002 [16]), and is introduced into §4.This theory is reorganized to be applicable to the core–periphery model in §5.
8See §5.4 for the precise meaning of this figure.9See, e.g., Crisfield (1977) [5] for an explanation of this theory.
10The group-theoretic bifurcation theory represents a map and the computational bi-furcation theory represents a car in the tracing of complex equilibria. See, e.g., Ikeda andMurota, 2002 [16] for successful combinatory use of these two theories.
4
system of cities, for the same value of transport cost, has increasing numbers
of stable equilibria when the number of cities increases. Such an increase
of stable equilibria is a fundamental difficulty that might instill pessimism
about the usefulness of the bifurcation analysis of the racetrack economy.
Nonetheless, as the most likely course of agglomerations of a system of cities,
the spatial period doubling cascade for 4, 8, and 16 cities is actually found
through numerical simulation, and thereby supports that pessimistic view.
This paper is organized as follows. The core–periphery model is intro-
duced into §2. Governing equations are presented with a study of stability
in §3. A brief explanation of group-theoretic bifurcation theory is offered in
§4. Bifurcation theory of the racetrack economy is described in §5. Agglom-
erations of the racetrack economy with a system of cities are investigated in
§6. The Appendix offers theoretical details and proofs.
5
2 Core–periphery model
A core–periphery model with an arbitrary discrete number of cities is pre-
sented as a recapitulation and a reorganization of Krugman (1991) [18] and
Fujita et al. (1999, Chapter 5) [10]. The economy comprises n possible
locations (labeled i = 1, . . . , n) around a circumference of a racetrack, two
industrial sectors (agriculture and manufacture), and two factors of produc-
tion (agricultural labor and manufacturing labor). The agricultural sector is
perfectly competitive and produces a homogeneous good, whereas the manu-
facturing sector is imperfectly competitive with increasing returns, producing
various and differentiated goods. Manufacturing laborers are mobile across
locations, but agricultural laborers are immobile. Laborers of each type
consume two goods and supply one unit of labor inelastically. The utility
functions are identical for agricultural labor and manufacturing labor. The
equilibrium of the model is determined through two stages: the short-run
equilibrium and the long-run equilibrium. The short-run equilibrium is de-
termined according to the spatial allocation of manufacturing workers. In
the long-run equilibrium, manufacturing workers can migrate to a city with
a higher real wage. As a result of such manufacturing workers’ migration,
the spatial allocation of manufacturing workers is determined.
2.1 Short-run equilibrium
The short-run equilibrium determines the income of each city, the price index
of manufactures in that city, the wage rate of workers in that city, and the real
wage rate in that city given the spatial allocation of manufacturing laborers
determined by the long-run equilibrium.
The nominal wage rate wi for the manufacturing labor force of the ith
city is given as
wi = [n∑
j=1
Yjt1−σij Gσ−1
j ]1/σ, (i = 1, . . . , n); (1)
the manufactured price index for the ith city is given as
Gi = [n∑
j=1
λj(wjtij)1−σ]1/(1−σ), (i = 1, . . . , n). (2)
6
Here Yi signifies the total income for the ith city, tij denotes the transport
cost in terms of the amount of the manufactured good dispatched per unit
received, σ stands for the elasticity of substitution between differentiated
goods, Gi denotes the manufactured price index, and λi (i = 1, . . . , n) stands
for the ratio of the manufacturing labor force for the ith city to the whole
manufacturing force, which is called the population of the ith city for short.
The total income for the ith city is expressed as
Yi = µλiwi + (1 − µ)/n, (i = 1, . . . , n), (3)
assuming that the agricultural wage has unity as a numeraire, where µ is the
ratio of the manufacturing labor force, the first term µλiwi in the right-hand-
side of (3) is the income of the manufacturing labor force, and the second
term (1 − µ)/n is that of the agricultural one.
2.2 Long-run equilibrium
In the long-run equilibrium, the manufacturing workers are assumed to mi-
grate to a city with a higher real wage. We express the real wage ωi of the
ith city as
ωi = wiG−µi , (i = 1, . . . , n), (4)
and consider the highest equilibrium real wage ω.
The long-run equilibrium employs the complementarity condition
ωi − ω = 0, (λi > 0),ωi − ω ≤ 0, (λi = 0),
(5)
(i = 1, . . . , n) and the conservation law of population
n∑
i=1
λi = 1. (6)
2.3 Iceberg transport costs
We employ the iceberg form of transport costs and define them as follows.
7
Assumption 1 (Iceberg transport costs). For the racetrack economy on
a circle with the unit radius, which is studied in this paper, we define the
transport cost tij between the two cities i and j by
tij = exp(τDij), (i, j = 1, . . . , n), (7)
where τ is the transport parameter and
Dij =2π
nmin(|i − j|, n − |i − j|), (i, j = 1, . . . , n)
represents the shortest distance between cities i and j; min(·, ·) denotes the
smaller value of the variables in parentheses.
8
3 Governing equations and stability
We have presented a set of equations for the core–periphery model in §2.
From these equations, we derive a system of nonlinear governing equations
of the model and derive the stability condition of the solutions of the model in
a manner suitable for the theoretical analysis of the racetrack economy in §5and the numerical analysis in §6. In the derivation of the governing equations,
the condensation is conducted on the set of equations to suppress auxiliary
equations and variables in §3.1. The stability condition is formulated in terms
of the eigenanalysis of a Jacobian matrix of the governing equations in §3.2.
3.1 Governing equations
Among many variables and parameters of these equations, we regard λ =
(λ1, . . . , λn)⊤ as an independent variable vector and τ as a bifurcation pa-
rameter11, and condense12 other variables as below.
The real wage in (4) can be expressed from (1)–(3) and (7) as a function
of (λ, τ) as
ωi = ωi(λ, τ), (i = 1, . . . , n). (8)
The highest equilibrium real wage among the cities can be expressed from
(5), (6), and (8) as a function of (λ, τ) as
ω = ω(λ, τ). (9)
We express a system of governing equations for the model as
F (λ, τ) =
ω1(λ, τ) − ω(λ, τ)λ1
...ωn(λ, τ) − ω(λ, τ)λn
= 0, (10)
11µ and σ are regarded as auxiliary parameters that are pre-specified for each problem.12A numerical counterpart of the condensation of the variables that is used in the nu-
merical bifurcation analysis in §6 is presented in Appendix A.
9
G(λ, τ) =
−ω1(λ, τ) − ω(λ, τ)...
−ωn(λ, τ) − ω(λ, τ)λ1...
λn
≥ 0 (11)
from (5) with (8) and (9).
Assumption 2 (Smoothness). F and G are sufficiently smooth functions.
Remark 1 The equality (10) is formulated as a special form in that λi is
multiplied to the ith component ωi(λ, τ) − ω(λ, τ) of (5). This special form
is pertinent to the discussion of asymptotic stability in §3.2 and of the study
of bifurcation in Appendix D.
Although there are variety of strategies13 to solve the equality (10) and
inequality (11) simultaneously, in favor of the consistency with bifurcation
theory, we use the following two-step strategy.
• Step 1: Obtain a family of solutions (equilibrium points) (λ, τ) of (10)
that forms smooth equilibrium path(s). A nonlinear system under-
going bifurcation involves several sets of equilibrium paths, including
bifurcated paths.
• Step 2: Among the equilibrium paths, we extract only those satisfying
the inequality (11), i.e., sustainable solutions (ωi(λ, τ) − ω(λ, τ) ≤ 0)
with non-negative populations (λi ≥ 0) (i = 1, . . . , n).
13The variational inequality approach, for example, is known as a strategy to tacklesuch problems (e.g. Nagurney, 1993 [23]; Facchinei and Pang, 2003 [7]).
10
3.2 Stability and economical feasibility of solutions
We introduce a local stability condition14 based on the dynamics15
dλ
dt= F (λ, τ). (12)
If all eigenvalues ei (i = 1, · · · , n) of the Jacobian matrix of F ,
J(λ, τ) =∂F
∂λ,
have negative real parts at a solution (λ, τ), then the solution is linearly
stable, and is asymptotically stable as t → ∞. If at least one eigenvalue has a
positive real part, then the solution is linearly unstable, and is asymptotically
unstable as t → ∞.
In practice, we are interested in solutions that are stable and satisfy
the inequality (11) and call such solutions (economically) feasible solutions.
Solutions which are not feasible are called (economically) infeasible solutions,
which include:
• unstable solutions for which ei for some i has a positive real part,
• solutions with negative population λi < 0 for some i, and/or
• unsustainable solutions with ωi − ω > 0 (λi = 0) for some i.
Proposition 1 below is pertinent in the check of feasibility.
Proposition 1 The feasibility of a solution (λ, τ) that satisfies the equal-
ity condition (10) with non-negative populations λi ≥ 0 (i = 1, . . . , n) is
classifiable as follows:
i) The solution is feasible if all eigenvalues ei (i = 1, . . . , n) of J(λ, τ)
have negative real parts.
ii) The solution is infeasible if an eigenvalue(s) has a positive real part(s).
Proof. See Appendix C. ¤
14The present stability condition is based on the asymptotic stability (e.g., Lorenz,1997 [21]; Ikeda and Murota, 2002 [16]).
15The dynamics in (12) corresponds to the replicator dynamics (cf. Fujita et al. 1999,[10]).
11
4 Group-theoretic bifurcation theory
The break bifurcation16 is explained in light of group-theoretic bifurcation
theory. This theory, a standard means to describe the bifurcation of symmet-
ric systems, has been developed to obtain the rules of pattern formation—
emergence of solutions with reduced symmetries via so-called symmetry-
breaking bifurcations (cf. Golubitsky et al., 1988 [14]). This theory will
be employed to investigate possible bifurcations of the racetrack economy in
§5.
We are interested in a symmetric system that satisfies the symmetry
condition, called the equivariance17
T (g)F (λ, τ) = F (T (g)λ, τ), g ∈ G (13)
of F (λ, τ) to a symmetry group G in terms of an n × n orthogonal matrix
representation T (g) of G that expresses the geometrical transformation for
an element g of G.
The Jacobian matrix J(λ, τ) is endowed with the symmetry condition
T (g)J(λ, τ) = J(λ, τ)T (g), g ∈ G (14)
if λ is G-symmetric in the sense that T (g)λ = λ (g ∈ G). By virtue of (14),
it is possible to construct a transformation matrix H, the column vectors
of which are made up of discrete Fourier series (cf. Murota and Ikeda, 1991
[22]), such that the Jacobian matrix J is transformed into a block-diagonal
form:
J = H⊤JH =
J0 O
J1
O. . .
(15)
with diagonal block matrices Jk (k = 0, 1, . . .). A diagonal block, say J0,
has G-symmetric eigenvectors, while eigenvectors of other blocks Jk (k =
1, 2, . . .) have reduced symmetries labeled by subgroups Gk (k = 1, 2, . . .)
16See Appendix B.4 for the explanation of break bifurcation.17The equivariance (13) is not an artificial condition for mathematical convenience, but
a natural consequence of the objectivity of the equation: the observer-independence of themathematical description.
12
of G. This is a mechanism to engender symmetry breaking via bifurcation.
This block-diagonal form is suitable for an analytical eigenanalysis of the
Jacobian matrix (cf. §5.3).
The bifurcation of a symmetric system with the equivariance (13) has
been studied in group-theoretic bifurcation theory and has properties (cf. Ikeda
and Murota, 2002 [16]):
• Property 1: The symmetry of the equilibrium points is preserved until
branching into a bifurcated path.
• Property 2: In association with repeated bifurcations, one can find a
hierarchy of subgroups
G −→ G1 −→ G2 −→ · · · (16)
that characterizes the hierarchical change of symmetries. Here −→denotes the occurrence of break bifurcation.
• Property 3: A bifurcated path sometimes regains symmetry on a bifur-
cation point on another equilibrium path with a higher symmetry.
In this section, the bifurcation rule is described in such a sequence that the
symmetry is reduced successively via bifurcations. However, when we observe
some economic system by decreasing the transport cost monotonously from
∞ to 0, a bifurcated path sometimes regains symmetry at a bifurcation point
as explained in Property 3 presented above.
13
5 Bifurcation of a racetrack economy
The tomahawk bifurcation of Krugman’s core–periphery model with two
cities is well known to produce spontaneous concentration to a single city.
In contrast, it will be demonstrated in §6 that the racetrack economy of a
system of cities displays more complex bifurcation. The objective of this sec-
tion is to investigate such bifurcation by group-theoretic bifurcation theory
presented in §4. We present several theoretical developments that will be
employed in the analysis of the racetrack economy in §6:
• Symmetry of the racetrack economy and its governing equation are
illustrated in §5.1.
• Trivial solutions18 of the racetrack economy are determined in view of
the symmetry in §5.2.
• Possible bifurcated solutions and possible courses of bifurcations from
the uniform population solution are investigated in §5.3.
• Among many possible equilibria predicted by the group-theoretic bi-
furcation theory, a spatial period-doubling cascade is advanced as the
most likely course en route to concentration in one city in §5.4.
• A systematic procedure to obtain equilibrium paths of the core–periphery
model is presented in §5.5.
5.1 Symmetry of racetrack economy and governing equa-
tion
We consider the racetrack economy with n cities that are equally spread
around the circumference of a circle as shown in Fig. 2, and describe the
symmetry of these cities and of the governing equation.
Assumption 3 (Parity). We set n to be even as the number of cities treated
in the numerical analysis is n = 4, 6, 8, and 16 (cf. §6).
18The trivial solutions denote solutions for which the population λ of the cities remainsunchanged in association with the change of the transport parameter τ (cf. Appendix B.2).
14
2
3
1
n
1-n
Figure 2: Racetrack among n cities (n = 16)
2
3 1
4
2
3
1
4
2
31
4
2
3
1
4
e = c(0) c(π/2) c(π) c(3π/2)
2
31
4
2
3
1
4
2
31
4
2
3
1
4
σc(0) σc(π/2) σc(π) σc(3π/2)
Figure 3: Actions of elements of D4
5.1.1 Groups for expressing symmetries
The symmetry of these cities can be described as the invariance under geo-
metrical transformations by the dihedral group G = Dn of degree n express-
ing regular n-gonal symmetry. This group is defined as
Dn = c(2πi/n), σc(2πi/n) | i = 0, 1, . . . , n − 1,
where · denotes a group consisting of the geometrical transformations in the
parentheses, c(2πi/n) denotes a counterclockwise rotation about the center
of the circle at an angle of 2πi/n (i = 0, 1, . . . , n − 1), and σc(2πi/n) is
the combined action of the rotation c(2πi/n) followed by the upside-down
reflection σ (cf. Fig. 3 for n = 4).
Bifurcated solutions from the Dn-symmetric racetrack economy have re-
15
D4 D2
D1
D2,41 D3,4
1 D4,41 C1
Figure 4: Symmetries of solutions for the four cities (n = 4; dashed line, axisof reflection symmetry; the area of ⃝ denotes the size of population)
duced symmetries that are labeled by subgroups19 of Dn. These subgroups
are dihedral and cyclic groups that are given respectively as
Dk,nm = c(2πi/m), σc(2π[(k − 1)/n + i/m]) | i = 0, 1, . . . ,m − 1,Cm = c(2πi/m) | i = 0, 1, . . . ,m − 1.
Therein, the subscript m (= 1, . . . , n/2) is an integer that divides n; the
superscript k (= 1, . . . , n/m) expresses the directions of the reflection axes.
Cm denotes cyclic symmetry at an angle of 2π/m; Dk,nm denotes reflection
symmetry with respect to m-axes together with this cyclic symmetry.
In general, spatial distribution of populations with a higher symmetry
with a larger m represents a more uniform state, while that with a lower
symmetry with a smaller m represents a more concentrated state.
Example 1 The symmetries of solutions, for example, for the four cities
(n = 4) are classified by these groups in Fig. 4. The patterns associated with
the groups D2,41 and D4,4
1 have the same economic meaning; such is also the
case for the groups D1 and D3,41 . ¤
19D2,nn/2
-, Cn-, and Cn/2-symmetric modes are absent for this specific racetrack problem.
16
5.1.2 Spatial periods
In the interpretation of agglomeration patterns, we consider the spatial pe-
riod T along the unit circle of the racetrack economy. When the cities are
invariant under the transformation c(2πi/m), i.e., Dm- or Dk,nm -invariant, we
define the spatial period as
T = Tm = 2π/m.
Likewise we define the spatial period for the Dn-invariant cities as
T = Tn = 2π/n.
In general, a higher spatial period with a larger m represents a more dis-
tributed spatial distribution of populations, a lower spatial period with a
smaller m represents a more concentrated distribution.
5.1.3 Equivariance
In the description of the bifurcation of the racetrack economy, the following
lemma for the equivariance of this economy is important as it paves the way
for application of the group-theoretic bifurcation theory in §4 for a particular
case of G = Dn. The rule of hierarchical bifurcations in (16) for G = Dn
varies according to the value of the integer n (cf. Appendix D.1).
Lemma 1 The nonlinear equality equation F (λ, τ) = 0 in (10) of the race-
track economy is endowed with equivariance with respect to Dn:
T (g)F (λ, τ) = F (T (g)λ, τ), g ∈ Dn. (17)
Therein T (g) simultaneously permutes the order of equations via T (g)F and
the order of independent variables via T (g)λ.
Proof. See Appendix C. ¤
17
5.2 Determination of trivial solutions
The symmetry of the racetrack economy engenders trivial solutions (cf. Ap-
pendix B.2), which satisfy, for any values of τ , the nonlinear equality equation
F (λ, τ) = 0 in (10). It is readily apparent that the racetrack economy has
the uniform-population trivial solution
λ = (1/n, . . . , 1/n)⊤ (18)
with Dn-symmetry and the spatial period of Tn = 2π/n.
In addition to the uniform population solution in (18), several trivial
solutions exist as expounded in Proposition 2.
Proposition 2 (Period multiplying trivial solutions). There are n/m trivial
solutions with
λi =
1/m, (i = k, k + n/m, . . . , k + (m − 1)n/m),0, otherwise
(19)
(m divides n; k = 1, . . . , n/m), which, for example, for k = 1 is Dm-
symmetric. The spatial period Tm = 2π/m of these solutions along the circle
becomes (n/m)-times as long as the period Tn = 2π/n of the uniform popu-
lation solution in (18).
Proof. See Appendix C. ¤
Among the trivial solutions in Proposition 2, we are particularly inter-
ested in the following trivial solutions:
• Period-doubling trivial solutions
λ = (2/n, 0, . . . , 2/n, 0)⊤ and (0, 2/n, . . . , 0, 2/n)⊤ (20)
express Dn/2-symmetric solutions, for which a concentrating city and
an extinguishing city alternate along the circle. The spatial period
Tn/2 = π/n of these solutions along the circle is doubled in comparison
with the period Tn = 2π/n of the uniform population solution in (18).
Note that the two solutions in (20) have the same economical meaning.
18
Table 1: Examples of trivial and non-trivial solutions (dashed line, axis ofreflection symmetry)
(a) Two cities (n = 2)
D2 D1
Trivial
Non-trivial
(b) Four cities (n = 2)
D4 D2 D1 D2,41 C1
Trivial Non-existent
Non-trivial Non-existent
• Concentrated trivial solutions
λ =( i
0, . . . , 0, 1, 0, . . . , 0)⊤, (i = 1, . . . , n)
express the concentration of the population to a single city. The solu-
tion for i = 1, for example, is D1-symmetric.
The variety of trivial solutions becomes diverse as the number n of cities
increases; see examples below.
Example 2 Two cities (n = 2) have only two trivial solutions20 as shown in
Table 1(a):
20These trivial solutions were observed by Krugman (1991) [18].
19
• the uniform population solution λ = (1/2, 1/2)⊤ (D2-symmetry), and
• the two concentrated solution λ = (1, 0)⊤ and λ = (0, 1)⊤ (D1-symmetry),
which are also the period-doubling trivial solutions.
Example 3 Four cities (n = 4) have more trivial solutions (cf. Table 1(b)),
such as
• the uniform population solution λ = (1/4, 1/4, 1/4, 1/4)⊤ (D4-symmetry),
• the period-doubling trivial solution λ = (1/2, 0, 1/2, 0)⊤ (D2-symmetry),
• the concentrated trivial solution λ = (1, 0, 0, 0)⊤ (D1-symmetry), and
• D2,41 -symmetric trivial solution λ = (0, 1/2, 1/2, 0)⊤, and so on. ¤
5.3 Bifurcation from the uniform population solution
Bifurcation from the Dn-symmetric uniform population solution in (18) is
investigated. Recall that n is assumed to be even.
According to the symmetry conditions (14), the Jacobian matrix J is a
symmetric circulant matrix with entries
Jij = kl, (l = min|i − j|, n − |i − j|)
for some kl (l = 1, 2, . . .).
The transformation matrix H for block-diagonalization in (15) is obtain-
able using the formula in Murota and Ikeda (1991) [22] as
H =
(η(+),η(−)), for n = 2,(η(+),η(−),η(1),1,η(1),2, . . . , η(n/2−1),1,η(n/2−1),2), for n ≥ 4.
(21)
The column vectors of this matrix H, which will turn out to be the eigen-
vectors of J , are expressed as the discrete Fourier series
η(+) =1√n
1...1
, η(−) =1√n
cos π · 0
...cos(π(n − 1))
=1√n
1−1...1
−1
, (22)
20
η(j),1 =
√2
n
cos(2πj · 0/n)
...cos(2πj(n − 1)/n)
, η(j),2 =
√2
n
sin(2πj · 0/n)
...sin(2πj(n − 1)/n)
,
(j = 1, . . . , n/2 − 1). (23)
These eigenvectors η(+), η(−), η(j),1, and η(j),2 are Dn-, Dn/2-, D1,nn/bn-, and
D1+bn/2,nn/bn -symmetric, respectively, and have the spatial periods of Tn = 2π/n,
Tn/2 = π/n, Tn/bn = nπ/n, and Tn/bn = nπ/n, respectively. Here
n = n/gcd (j, n) ≥ 3, (24)
and gcd (j, n) is the greatest common divisor of j and n.
The block-diagonal form in (15) reduces to a diagonal matrix as
J = H⊤JH = diag(e(+), e(−), e(1), e(1), . . . , e(n/2−1), e(n/2−1)),
where diag(·) denotes a diagonal matrix with the diagonal entries therein.
The diagonal entries, which correspond to the eigenvalues of J , are
e(+) = k0 + kn/2 + 2
n/2−1∑
l=1
kl, (25)
e(−) = k0 + (−1)n/2kn/2 + 2
n/2−1∑
l=1
(−1)lkl, (26)
e(j) = k0 + cos(πj) kn/2 + 2
n/2−1∑
l=1
cos(2πjl/n) kl, (j = 1, . . . , n/2 − 1). (27)
It is noteworthy that e(+) and e(−) are simple eigenvalues, and that e(j) (j =
1, . . . , n/2 − 1) are double eigenvalues that are repeated twice.
We can classify critical points on the uniform population solution as
e(+) = 0 : limit point of τ (M = 1),e(−) = 0 : simple bifurcation point (M = 1),e(j) = 0 : double bifurcation point (M = 2).
These simple and double bifurcation points are break bifurcation points.
The simple bifurcation point with e(−) = 0 corresponds to the spa-
tial period-doubling bifurcation, which engenders an alternating equilibrium
(cf. Proposition 3).
21
Proposition 3 (Simple bifurcation). At the simple bifurcation point, which
is either a pitchfork or tomahawk, we encounter a symmetry-breaking bi-
furcation Dn −→ Dn/2. Its critical eigenvector is given uniquely as a Dn/2-
symmetric vector η(−) of (22) with components of alternating signs expressing
the bifurcation mode of spatial period doubling from T = 2π/n to π/n.
The double bifurcation point e(j) = 0 for some j corresponds to the spatial
period n-times bifurcation (cf. (24) for the definition of n), which engenders a
more rapid concentration than the period doubling bifurcation of the simple
bifurcation point (cf. Proposition 4). Double bifurcation points with e(j) = 0
are absent for the two cities with n = 2 (cf. (21)). It exercises caution
that the bifurcation of the two cities is a special case, while four or more
cities in general have double bifurcation points and have different bifurcation
properties.
Proposition 4 (Double bifurcation). At the double bifurcation point with
e(j) = 0 for some j, we encounter a symmetry-breaking bifurcation Dn −→Dn/bn, at which the spatial period becomes n-times ( n ≥ 3 by (24)).
Proof. See Appendix C. ¤
Example 4 The change of symmetry at bifurcation points is illustrated in
Fig. 5 for the four cities (n = 4). At the simple bifurcation point in Fig. 5(a),
the bifurcation doubles the spatial period and triggers concentration of the
population to two cities located at opposite sides of the circle, while the
populations of the other two cities decline. At the double bifurcation point
in Fig. 5(b) associated with e(j) = 0 (n = 4, j = 1, n = 4), the spatial period
becomes four times. ¤
5.4 Spatial period-doubling cascade
In addition to break bifurcations from the uniform-population trivial solu-
tion with Dn-symmetry that were studied in §5.3, several possible sources
of symmetry-breaking exist. Namely, further break bifurcations may be en-
countered on (a) bifurcated paths of this Dn-symmetric solution and (b)
Dm-symmetric trivial solutions (m divides n) presented in §5.2.
22
D4 D2
(a) Simple bifurcation
D3,4
1D4 D1 D2,4
1 D4,4
1
(b) Double bifurcation
Figure 5: Direct bifurcations from the four uniform cities (n = 4; the arrowdenotes the occurrence of a bifurcation)
D4 D1D2D8
Figure 6: Spatial period-doubling cascade for the eight cities (n = 8; thearrow denotes the occurrence of a bifurcation)
All these break bifurcations can be described by group-theoretic bifurca-
tion theory (cf. Ikeda and Murota, 2002 [16]). The rule of bifurcation depends
on the integer number n, to be precise, the divisors of the number n. The
bifurcation becomes increasingly hierarchical and complex for n with more
divisors (cf. Appendix D).
Among possible courses of hierarchical bifurcations, we pay special atten-
tion to the spatial period-doubling bifurcation for Dn-symmetric cities with
n = 2k (k is some positive integer) that is expounded in Proposition 5 below.
Figure 6 depicts this bifurcation for n = 8 = 23 cities.
Proposition 5 (Spatial period-doubling cascade). Dn-symmetric cities with
23
n = 2k for some integer k have a possible course:
D2k −→ D2k−1 −→ D2k−2 −→ · · · −→ D1 (28)
that is called “spatial period-doubling cascade21,” in which the spatial period
is doubled successively by repeated simple bifurcations.
Remark 2 Proposition 5 serves as a generalization of the study of Tabuchi
and Thisse (2009) [27] who conducted a local analysis (linearized eigenprob-
lem) for the flat distribution of the racetrack economy to predict of the
occurrence of the period doubling cascade. In comparison with the study
by Tabuchi and Thisse (2009) [27], the implementation of the income effect
for good consumption of the core–periphery model, i.e., an increase in goods
consumption in association with an increase in the income, is a possible im-
provement of this paper from an economics standpoint.
5.5 Systematic procedure to obtain equilibrium paths
We present a systematic procedure to obtain equilibrium paths of the core–
periphery model. First, we conduct the exhaustive search by obtaining all
the equilibrium paths using the following steps:
Step 1: Obtain all trivial solutions by the method presented in §5.2.
Step 2: Carry out the eigenanalysis of the Jacobian matrix J , on these trivial
solutions to obtain the bifurcation points and to classify feasible and
infeasible solutions. On the uniform population solution, the formulas
(25)–(27), which give the eigenvalues analytically, are to be employed,
while the numerical eigenanalysis is to be conducted for other trivial
solutions.
Step 3: Obtain the bifurcated paths branching from all these trivial solutions
by the computational bifurcation theory in Appendix E. The numerical
21A repeated doubling of time period by bifurcations takes place in many physical sys-tems (Feigenbaum, 1978 [8]) and is called period doubling cascade.
24
eigenanalysis is to be conducted to find critical points and testify the
feasibility of these solutions.
Step 4: Repeat the Steps 3 and 4 to exhaust all equilibrium paths.
Next, among all these equilibrium paths we select feasible ones that are
to be encountered when the transport cost τ is decreased from ∞ to 0.
Existence and multiplicity of possible feasible ones for a particular value of τ
depend on the umber n of cities, the values of the parameters σ and µ, and
so on, and must be investigated individually.
25
6 Bifurcation analysis of racetrack economy
Agglomerations of the racetrack economy are investigated for n = 4, 6, 8, and
16 cities. The solutions of a system of governing equations (10) and (11) of
the core–periphery model are obtained by the systematic procedure to obtain
equilibrium paths in §5.5. The progress of agglomeration is expressed as
successive breaking of symmetries associated with successive elongation of the
spatial period with reference to the theoretical rule of bifurcation presented in
§5. Successive and gradual progress of agglomerations by the spatial period-
doubling cascade in Proposition 5 is highlighted as a key phenomenon for
n = 4, 8, and 16 cities in §6.1. The period-doubling and period-tripling are
observed for n = 6 cities in §6.2.
We set the elasticity of substitution as σ = 10.0 and the ratio of the
manufacturing labor force as µ = 0.4. The transport parameter τ , which is
proportional to the transport cost tij via (7) and stays in the range [0,∞], is
scaled as
τ ′ = 1 − exp(−τπ); (29)
τ ′ = 0 (τ = 0) corresponds to the state of no transport cost, and τ ′ = 1
(τ = +∞) corresponds to the state of infinite transport cost.
6.1 Period Doubling Cascade
We demonstrate the occurrence of period doubling cascade for n = 4, 8, and
16 cities.
6.1.1 Four cities
For the four cities (n = 4), equilibrium paths were obtained by the by the
systematic procedure to obtain equilibrium paths in §5.5. Figure 7(a) shows
τ ′ versus λ1 curves obtained in this manner, where the ordinate τ ′ = 1 −exp(−τπ) is a scaled transport parameter in (29). Economically feasible
solutions (shown as solid curves) and infeasible ones (as dotted curves) are
classified using Proposition 1 in §3.2. Trivial paths (solutions) with Dm-
symmetries (m = 1, 2, 4) exist at the horizontal lines at λ1 = 0, 1/4, 1/2,
26
0
0.5
1
λ1
0.5 10
τ
0.5 1
0
0.5
1
λ1
OA
FE
D
C B
D1
D4
D2
D2
0
D1
D1
D
E F
τ
(a) All equilibrium paths (b) Feasible trivial pathsand associated paths
Simple
bifurcation CDynamical
shift
OA
Dynamical
shift
EFCDBC
(c) Predicted shift of feasible solutions as τ ′ decreases
Figure 7: Equilibrium paths of the four cities (n = 4) and a predicted shiftof feasible solutions in association with the decrease of transport cost (τ ′ =1− exp(−τπ); solid curve, feasible; dashed curve, infeasible; , simple breakpoint; , limit point; •, sustain point)
27
and 1 (cf. Table 1(b) in §5.2), and several bifurcated paths connecting these
trivial paths exist. These paths are apparently quite complex.
To assist the economical interpretation, among such complex paths, we
have chosen feasible trivial paths and associated paths shown in Fig. 7(b)
as those most likely to occur; distributions of populations are portrayed at
several equilibrium points. Economically feasible parts (shown as solid lines)
of the trivial solutions are
• OA: uniform population solution λ = (1/4, 1/4, 1/4, 1/4)⊤ (D4-symmetry),
• BC: period-doubling solution λ = (1/2, 0, 1/2, 0)⊤ (D2-symmetry),
• EF: concentrated solution λ = (1, 0, 0, 0)⊤ (D1-symmetry), and
• E′F′: another concentrated solution λ = (0, 0, 1, 0)⊤ (D1-symmetry).
Note that EF and E′F′ are symmetric counterparts with the same economical
meaning.
Bifurcation points on these trivial solutions are classifiable as break and
sustain points (cf. Appendix B.4). Symmetries of the system are reduced at
period-doubling breaking bifurcation points A and C denoted as (cf. Ap-
pendix D.1):
• At A, we encounter a symmetry breaking D4 −→ D2 associated with
λ = (1/4, 1/4, 1/4, 1/4)⊤ → (1/4 + α, 1/4 − α, 1/4 + α, 1/4 − α)⊤,
(|α| < 1/4).
• At C, we encounter a symmetry breaking D2 −→ D1 associated with
λ = (1/2, 0, 1/2, 0)⊤ → (1/2 + α, 0, 1/2 − α, 0)⊤, (|α| < 1/2).
Symmetries are preserved at the sustain points denoted as •, at which a
trivial solution and a non-trivial one intersect (Appendix D). Sustain point
B has D2-symmetry; E and E′, D1-symmetry.
Among the bifurcated paths, we found the path CD and its symmetric
counterpart CD′ feasible. The feasible path CD became infeasible at the
28
limit (maximum) point τ at D denoted by (cf. the left of Fig. 11(a) in
Appendix B.3).
In view of the whole set of feasible paths obtained herein, in association
with the decrease of τ ′, we predict a possible course of the accumulation of
population following four feasible stages: OA, BC, CD, and EF, as presented
in Fig. 7(c). Dynamical shifts are assumed between OA and BC and between
CD and EF. Starting from the uniform state λ = (1/4, 1/4, 1/4, 1/4)⊤, via
bifurcations and dynamical shifts, we arrive at the complete concentration
λ = (1, 0, 0, 0)⊤, in agreement with the rule of bifurcations in Fig. 13(a)
in Appendix D.1. We can see the occurrence of a spatial period-doubling
cascade
D4 −→ D2 −→ D1,
en route to the concentration to a single city, in agreement with Proposition 5
in §5.4.
Recall that the feasible solutions of the simple tomahawk bifurcation22 of
the two cities consisted only of two trivial solutions: the uniform population
solution and the completely concentrated solution (cf. Table 1(a)). Differ-
ent from the two cities, the four cities have a feasible non-trivial solution23
CD, for which migration from one city to another occurs in an economically
feasible manner without undergoing bifurcation. Moreover, the progress of
agglomeration of the four cities is much more complex than that of the spon-
taneous concentration of the two cities triggered by the simple tomahawk
bifurcation. It demands caution that the experience of the two cities is not
universal, thereby underscoring the importance of bifurcation analysis for
many cities examined in the remainder of this section.
6.1.2 Eight cities
For the eight cities bifurcated paths branching from several trivial solutions
are obtained in an exhaustive manner as shown in τ ′ versus λ1 relationship
22The tomahawk bifurcation was observed, e.g., in Krugman (1991) [18] and Fujitaet al. (1999) [10] for the present model, and in Forslid and Ottaviano (2003) [9] for ananalytically solvable model.
23A feasible non-trivial solution was observed also by Pfluger (2004) [25] for a simple,analytically-solvable model for the two cities.
29
0 0.5 1
0
0.5
1
1λ
τ0 0.5 1
0
0.5
1
1λ
D8
D4
D1
D1
D4
D2
D2
τ
(a) All equilibrium paths (b) Feasible trivial pathsand associated paths
Figure 8: Equilibrium paths of the eight cities expressed in terms of τ ′ versusλ1 curves (n = 8; τ ′ = 1 − exp(−τπ); solid curve, feasible; dashed curve,infeasible)
0 0.5 1
0
0.5
1
1λ
D1
D1
D2
D2
D4
D4
D8
D16
D8τ
Figure 9: Feasible trivial paths and associated paths for the 16 cities thatare expected to be followed in association with a decrease of τ ′ (n = 16;τ ′ = 1 − exp(−τπ); solid curve, feasible; dashed curve, infeasible)
30
of Fig. 8(a). The horizontal lines at λ1 = 0, 1/8, 1/4, 1/2, and 1 are trivial
solutions with Dm-symmetries (m = 1, 2, 4, 8); these bifurcated paths that
connect these trivial solutions have grown more complex than those for the
six cities in Fig. 10(a).
Among all the equilibrium paths for the eight cities shown in Fig. 8(a),
feasible equilibrium paths that are expected to be followed in association
with the decrease of τ ′ are depicted in Fig. 8(b). The spatial period-doubling
cascade
D8 −→ D4 −→ D2 −→ D1 (30)
engenders concentration into four cities and then into two cities, en route to
concentration to a single city.
Complex bifurcated paths connecting these trivial solutions were found
in Fig. 8(a). Such complexity, notwithstanding, all these paths have been
traced successfully by the systematic procedure to obtain equilibrium paths
in §5.5; it demonstrates the prowess of this procedure. In addition, the rule
of break bifurcations in Fig. 13(b) in Appendix D.1 was of assistance in
the tracing of bifurcated paths. One might feel pessimistic when observing
the complexity of the bifurcation of the racetrack economy that will grow
rapidly with the increase of the number n of cities. Nonetheless, we can
resolve such pessimism by addressing only feasible solutions, as in the spatial
period-doubling cascade (30).
6.1.3 16 cities
Similarly to the four and eight cities, the 16 cities displayed the spatial-period
doubling cascade, as shown in Fig. 9,
D16 −→ D8 −→ D4 −→ D2 −→ D1.
6.1.4 Discussion
The presence of spatial-period doubling cascade, which was predicted in
Tabuchi and Thisse (2009) [27] and also by group-theoretic bifurcation the-
ory in Proposition 5, has thus been ensured. It is highlighted as a mechanism
to engender concentration out of uniformity, especially for n = 2k cities. It is
31
to be remarked again that, unlike the two-city special case, there are feasible
non-trivial solutions.
6.2 Period doubling and tripling: six cities
Equilibrium paths of the six cities are shown in Fig. 10(a), from which we
chose feasible paths and some associated paths shown in Fig. 10(b).
There are trivial solutions with Dm-symmetries (m = 1, 2, 3, 6) (cf. §5.2)
at the horizontal lines at λ1 = 0, 1/6, 1/3, 1/4, 1/2, and 1:
• λ1 = 1/6: D6-symmetric uniform population solution,
• λ1 = 0, 1/3: D3-symmetric period-doubling trivial solutions,
• λ1 = 0, 1/4, 1/2: D2-symmetric trivial solutions, and
• λ1 = 0, 1: D1-symmetric concentrated trivial solutions.
We observed
• period doubling simple break bifurcations: D6 −→ D3 and D2 −→ D1;
• period tripling double break bifurcations: D6 −→ D2 and D3 −→ Dk,61 .
This is due to the fact that n = 6 has two divisors: 2 and 3. Thus the period
doubling is not that dominant as the four cities (cf. Fig. 7(b)), while the
period tripling via double break bifurcations plays an important role. The
period tripling, which is theoretically predicted in Proposition 2 with n = 3,
does not take place for n = 2k cities, including the two cities.
A predicted shift of feasible solutions occurring in association with the
decrease of τ ′ is presented in Fig. 10(c). This shift is not unique:
• The D6-symmetric state might dynamically jump into either the D2-
or D3-symmetric state.
• The D3-symmetric state might dynamically jump into the D2- or D5,61 -
symmetric state.
By virtue of the mixed occurrence of the period doubling and tripling, the
predicted shift for the six cities with n = 6 = 2 × 3 is more complex than
that of the four cities portrayed in Fig. 7(b).
32
0.5
0.5 0.75 1
0
1
1λ
τ
0.5
0.5 0.75 1
0
1
1λ
D6
τ
D5,6
1D1D11
D3
D3
D2
D1
C1
C1
D1
D1
D1
C1
D5,6
1D1D11
(a) All equilibrium paths (b) Feasible trivial pathsand associated paths
D5,61D1D11
D2
D6
Dynamical
shiftD3
Dynamical
shift
Simple
bifurcation
D1
C1
Dynamical
shift
Dynamical
shift
(c) Predicted shift of feasible solutions as τ ′ decreases
Figure 10: Equilibrium paths of the six cities (n = 6; τ ′ = 1 − exp(−τπ);solid curve, feasible; dashed curve, infeasible)
33
7 Conclusions
To testify the adequacy of the two-city special case as a platform for spa-
tial agglomerations, we investigated the progress of agglomerations of the
racetrack economy of the core–periphery model with four, six, eight, and 16
cities. These cities displayed several features, including:
• feasible non-trivial solutions,
• period-doubling cascade,
• a plethora of bifurcated paths, and
• period tripling via double break bifurcations.
These features were not observed and can be overlooked in Kruhgman’s two-
city special case, in which the tomahawk bifurcation engenders spontaneous
concentration to a single city. It demands caution that the experience of the
two-city special case with a simple tomahawk bifurcation is not universal. It
is preferable to employ a system of cities as a platform for the investigation
of spatial agglomerations.
Symmetry-breaking bifurcation predicted by group-theoretic bifurcation
theory proposed a broader view on the bifurcation of the racetrack economy.
In fact, the bifurcation phenomena become progressively complex in associa-
tion with the increase of the number of cities. Such complexity might instill
pessimism about the usefulness of the bifurcation analysis of the racetrack
economy. Yet, when we specifically examine economically feasible solutions
that are expected to occur in association with the decrease of the transport
cost, the spatial period-doubling cascade can be highlighted as the most likely
mechanism to engender concentration out of uniformly distributed popula-
tion. This suffices to clarify the pessimism.
Such complex phenomena can be traced in an exhaustive and systematic
manner owing to the insight of group-theoretic bifurcation theory. The pro-
posed procedure is applicable to any new economic geography models other
than Krugman’s core–periphery model, and also to city distributions other
than the racetrack economy. Accordingly, it will be a topic of future studies
34
to carry out bifurcation analysis of other updated new economic geography
models with a system of cities scattered on a two-dimensional domain.
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37
A Derivation of incremental equation
To derive the incremental equation for the equality (10), we rewrite the
complementary condition (5) as
P (λ, w, ω, τ) =
(ω1(w, τ) − ω)λ1
...(ωn(w, τ) − ω)λn
= 0, (A.1)
ωi(w, τ) − ω ≤ 0, λi ≥ 0, (i = 1, . . . , n). (A.2)
Here w = (w1, . . . , wn)⊤ and ωi = ωi(w, τ) (i = 1, . . . , n) by (4).
Equation (1) is expressed as
M (λ, w, τ) =
[ n∑s=1
Ys(t1s)1−σGσ−1
s
]−
(w1
)σ
...[ n∑
s=1
Ys(tns)1−σGσ−1
s
]−
(wn
)σ
= 0, (A.3)
and the conservation law (6) is (1 = (1, . . . , 1)⊤)
F (λ) = 1⊤λ − 1 = 0. (A.4)
An assembly of the relations (A.1), (A.3), and (A.4) gives the set of 2n+1
nonlinear equations
P (λ, w, ω, τ)M(λ,w, τ)
F (λ)
= 0 (A.5)
with a bifurcation parameter τ and 2n + 1 independent variables λ, w, and
ω.
We rewrite (A.5) into an incremental form:
∂P
∂λδλ +
∂P
∂wδw +
∂P
∂ωδω +
∂P
∂τδτ = 0, (A.6)
∂M
∂λδλ +
∂M
∂wδw +
∂M
∂τδτ = 0, (A.7)
∂F
∂λδλ = 0 (A.8)
38
(cf. Ikeda and Murota, 2002, Chapter 7 [16]), in which ∂F/∂λ = 1 by (A.4).
Under Assumption 4, it is possible to eliminate independent variables δw
and δω from (A.6)∼(A.8) as shown below:
δw =
(∂M
∂w
)−1(∂M
∂λJ−1∂F
∂τ− ∂M
∂τ
)δτ,
δω = −Tδτ,
and, in turn, to arrive at an incremental equilibrium equation
F (δλ, δτ) = Jδλ +∂F
∂τδτ + h.o.t. = 0, (A.9)
where h.o.t. dentoes higher order terms and
J =∂F
∂λ=
∂P
∂λ− ∂P
∂w
(∂M
∂w
)−1∂M
∂λ,
∂F
∂τ= S − T
∂P
∂ω,
S =∂P
∂τ− ∂P
∂w
(∂M
∂w
)−1∂M
∂τ, T =
∂F
∂λJ−1S
∂F
∂λJ−1
∂P
∂ω
.
Assumption 4 (Regularity conditions.) The incremental governing equa-
tion (A.9) was derived under the assumptions that the matrices J and ∂M/∂w
are nonsingular, and that the scalar ∂F/∂λJ−1∂P /∂ω is nonzero.
B Classifications and definition of equilibrium
points
In the study of the agglomeration of the core–periphery model, it is useful
to resort to various kinds of classifications of equilibrium points.
B.1 Interior and corner solutions
Solutions of this model are classifiable into two types:
39
• an interior solution for which all cities have positive population λi > 0
(i = 1, . . . , n), and
• a corner solution for which some cities have zero population.
The existence of the corner solution is a special feature of the core–periphery
model that demands reorganization in the application of bifurcation theory
(cf. Appendix D).
B.2 Trivial and non-trivial solutions
The core–periphery model has characteristic solutions, for which the popula-
tion λ of the cities remains unchanged in association with the change of the
transport parameter τ . We accordingly have classification:
Trivial solution: λ is constant with respect to τ .Non-trivial solution: λ is not constant with respect to τ .
B.3 Ordinary, limit, and bifurcation points
With reference to the eigenvalues ei (i = 1, . . . , n) of the Jacobian matrix J ,
equilibrium points are classified as
ei = 0 for all i, ordinary point,
ei = 0 for some i, critical (singular) point.
Critical points are classifiable as
limit point of τ (M = 1),
bifurcation point
simple (M = 1),double (M = 2),
...
where M is the multiplicity of a critical point that is defined as the number
of zero eigenvalues of J .
At a limit point of τ , as portrayed in Fig. 11(a), the value of τ is max-
imized or minimized. A feasible path is shown by the solid curve, and an
infeasible one by the dashed curve. Two half branches are connected at the
40
τ
λ i
τ
λ i
(a) Limit point of τ
τ
λ i
τ
λ i
τ
λ i
τ
λ i
(b) Bifurcation point
Figure 11: Critical points (solid curve, feasible; dashed curve, infeasible)
limit point; a part of the path beyond the limit point is called a half branch
and so is another part. With regard to economical feasibility, there are two
cases:
• A half branch is feasible, but another half branch is not.
• Both half branches are infeasible.
At a bifurcation point, two or more equilibrium paths intersect: Fig. 11(b)
presents a simple bifurcation point at which two paths (four half branches)
intersect. With regard to economical feasibility, there are four cases: Three,
two, one, or zero half branches are feasible, and the remaining half branches
are infeasible. If we focus only on feasible half branches, they look like a
two-pronged weapon, a curve with a kink, a branch, and so on.
B.4 Break and sustain points
Recall the block-diagonal form (15) in §4:
J = H⊤JH =
J0 O
J1
O. . .
.
41
τ
λ j
λ i
λ i
i
i
τ
λ iλ i
λ i
i
i
(a) Crossing point of two (b) Crossing point of trivialnon-trivial solutions and non-trivial solutions
Figure 12: Sustain points (solid curve, feasible; dashed curve, infeasible)
At a bifurcation point, a block Jk for some k becomes singular. Depending
on the type of block that becomes singular, bifurcation points are classified
into two types:
• A break bifurcation point, or a break point, is symmetry-breaking one,
at which Jk becomes singular for some k(≥ 1). The symmetry of the
system is reduced on a bifurcated path branching at a break point (cf.
§5).
• A sustain bifurcation point, or a sustain point is a symmetry-preserving
one, at which J0 becomes singular. The symmetry of the system is
preserved on a bifurcated path branching at a sustain point.
The sustain bifurcation point is an inherent feature of the present core–
periphery model that permits the extinction of city population of manu-
facturing labor. This point is necessarily a bifurcation point because the
factorized form (ωi − ω)λi of (10) (cf. Remark 1) produces two independent
solutions. The point, as shown in Fig. 12, is classified into two types24: (a)
the crossing point of two non-trivial solutions and (b) the crossing point of
a trivial solution and a non-trivial solution.
24The sustain point for the two cities in Fujita et al. (1999) [10] corresponds to thecrossing point of a trivial solution and a non-trivial solution in Fig. 12(b).
42
Since λi and ωi− ω vanish simultaneously at this point, the sign of ωi− ω
along a (trivial) solution path changes, as does the sign of λi along another
path. At the point, a sustainable solution (ωi−ω < 0, λi = 0) changes into an
unsustainable one (ωi− ω > 0, λi = 0) along a path, while a feasible solution
with positive population (λi > 0, ωi − ω = 0) changes into an infeasible one
with negative population (λi < 0, ωi − ω = 0).
Remark 3 Fujita et al. (1999) [10] considered only feasible solutions, and
regarded the sustain point as a kink that connect two half branches. Yet this
point is considered as a bifurcation point in this paper to be consistent with
the computational bifurcation theory in Appendix E.
C Proofs
Proof of Proposition 1. The Jacobian matrix of the equality condition
(10) reads
J =∂F
∂λ
=
ω1 − ω 0 · · · 0
0 ω2 − ω. . .
......
. . . . . . 00 · · · 0 ωn − ω
+
Ω11λ1 Ω12λ1 · · · Ω1nλ1
Ω21λ2 Ω22λ2. . .
......
. . . . . ....
Ωn1λn · · · · · · Ωnnλn
= diag(ω1 − ω, . . . , ωn − ω) + diag(λ1, . . . , λn)Ω, (C.1)
where diag(· · · ) denotes a diagonal matrix with diagonal entries therein and
Ωij =∂(ωi − ω)
∂λj
, (i, j = 1, . . . , n), (C.2)
Ω = (Ωij | i, j = 1, . . . , n). (C.3)
For an interior solution with ωi − ω = 0 and λi > 0 (i = 1, . . . , n) (cf. (5)
and Appendix B.1), the Jacobian matrix in (C.1) reduces to
J = diag(λ1, . . . , λn)Ω. (C.4)
43
An interior solution is stable if all the eigenvalues of the matrix Ω in (C.3)
have negative real parts (cf. §3.2) because the matrix diag(λ1, . . . , λn) in
(C.4) is positive definite.
A corner solution (cf. Appendix B.1) can be expressed without loss of
generality25 as
λi > 0, ωi − ω = 0, (i = 1, . . . ,m), (C.5)
λi = 0, (i = m + 1, . . . , n). (C.6)
With the use of (C.5), the Jacobian matrix in (C.1) becomes
J =
Φ1 Φ2
ωm+1 − ω 0
O. . .
0 ωn − ω
, (C.7)
where
Φi = diag(λ1, . . . , λm)Ωi, (i = 1, 2),
Ω1 =
Ω11 · · · Ω1m...
. . ....
Ωm1 · · · Ωmm
, Ω2 =
Ω1(1+m) · · · Ω1n
.... . .
...Ωm(1+m) · · · Ωmn
.
From (C.7), it is apparent that ei = ωi−ω (i = m+1, . . . , n) are eigenvalues of
J , whereas the other m eigenvalues ei (i = 1, . . . ,m) are given as eigenvalues
of Φ1.
For a stable corner solution, we have ei = ωi − ω < 0 (i = m + 1, . . . , n),
whereas ωi−ω = 0 (i = 1, . . . ,m) by (C.6). Therefore, the sustainability ωi−ω ≤ 0 (i = 1, . . . , n) in (5) is satisfied for the stable solution. Consequently,
the check of the sustainability is to be replaced with the investigation of
stability. ¤
25The consideration of this form does not lose generality because all corner solutions canbe reduced to the form by appropriately rearranging the order of independent variables λ.
44
Proof of Lemma 1. With the use of the representation matrices for
c(2π/n) and σ:
T (c(2π/n)) =
11
. . .
1
, T (σ) =
11
···1
,
the representation matrices T (g) (g ∈ Dn) can be generated as
T (c(2πi/n)) = T (c(2π/n))i, T (σc(2πi/n)) = T (σ)T (c(2π/n))i,
(i = 0, 1, . . . , n − 1).
In the proof of the equivariance (17), we note that
ω(T (g)λ, τ) = ω, ω(T (g)λ, τ) = T (g)ω(λ, τ),
where the former denotes the objectivity of ω with respect to the numbering
of cities, and the latter denotes that the rearrangement of λ leads to the
rearrangement of ω in the same order. Then, for example, for g = σ
F (T (σ)λ, τ) =
ω1(T (σ)λ, τ) − ω(T (σ)λ, τ)λ1
ω2(T (σ)λ, τ) − ω(T (σ)λ, τ)λn
ω3(T (σ)λ, τ) − ω(T (σ)λ, τ)λn−1...
ωn(T (σ)λ, τ) − ω(T (σ)λ, τ)λ2
=
ω1(λ, τ) − ω(λ, τ)λ1
ωn(λ, τ) − ω(λ, τ)λn
ωn−1(λ, τ) − ω(λ, τ)λn−1...
ω2(λ, τ) − ω(λ, τ)λ2
= T (σ)F (λ, τ).
This shows the equivariance (17) for g = σ. The equivariance for other
elements of g ∈ Dn can be shown similarly. ¤
45
Proof of Proposition 2. We consider Dm-symmetric state, for which
the equivariance (13) with G = Dm for the explicit form of F in (10) entails
ω1 = ω1+n/m = · · · = ω1+(m−1)n/m,ω2 = ω2+n/m = · · · = ω2+(m−1)n/m,
...ωn/m = ω2n/m = · · · = ωn.
(C.8)
As a candidate for a trivial solution, we consider a Dm-symmetric population
distribution
λi = 1/m, ωi − ω = 0, (i = 1, 1 + n/m, . . . , 1 + (m − 1)n/m),λi = 0, otherwise,
(C.9)
which is obtained by setting k = 1 in (19).
The substitution of (C.9) into (10) yields
F =
ω1 − ωλ1
...ωn − ωλn
=
0 × λ1
(ω2 − ω1) × 0...
(ωn/m − ω1) × 0...
= 0.
This proves that (C.9) is a trivial solution, while other trivial solutions are
treated similarly. ¤
Proof of Proposition 4. The critical eigenvector is given by the super-
position of the two vectors η(j),1 and η(j),2 in (23) as
η(θ) = cos θ · η(j),1 + sin θ · η(j),2
for general angle θ (0 ≤ θ < 2π). The bifurcated paths do not branch in the
general direction associated with arbitrary θ, but branch in finite directions
as expounded in Lemma 2. This is a novel aspect presented in this paper
that was not examined for the core–periphery model up to now.
Lemma 2 As made clear by group-theoretic analysis (cf. Ikeda et al., 1991 [17];
Ikeda and Murota, 2002 [16]), bifurcated paths branching at the double bifur-
cation point satisfy the following properties:
46
(i) There exist n bifurcated paths (2n half branches) in the directions of
δλ = Cη(αk), Cη(αk+bn), (k = 1, . . . , n),
where C is a scaling constant and
αi = −π (i − 1)/n, (i = 1, . . . , 2n).
(ii) The solutions δλ = Cη(αk) and Cη(αk+bn) are Dk,nn/bn-symmetric (k =
1, . . . , n). Therefore, the spatial period becomes n-times (n ≥ 3) in
comparison with that of the Dn-symmetric uniform population solution.
(iii) The 2n half branches are classifiable into two independent ones: δλ =
Cη(α2l−1), Cη(α2l) (l = 1, . . . , n). It suffices in numerical analysis to
find the two branches in two directions: δλ = Cη(α1), Cη(α2).
¤
Remark 4 Proposition 2 is extendible to double bifurcation points on bi-
furcated paths with Dm-symmetry (m divides n; m ≥ 3) by choosing η(j),1
and η(j),2 to be D1,mm/ bm- and D
1+ bm/2,mm/ bm -symmetric, respectively. Here m =
m/gcd (j,m) and 1 ≤ j < m/2.
D Hierarchical bifurcations
We explain the mechanism of hierarchical bifurcations of the racetrack econ-
omy that consist of symmetry-breaking at break points and the extinction
of city population of manufacturing labor at sustain points, en route to the
concentration of population in a city. As mentioned in Appendix B.4, these
points are characterized by
break point: symmetry breaking,sustain point: symmetry preserving.
47
D.1 Break bifurcations
In addition to break bifurcations from the uniform-population trivial solu-
tion with Dn-symmetry that were studied in §5.3, there are several possible
sources of symmetry-breaking. Namely, further break bifurcations may be
encountered on (a) bifurcated paths of this Dn-symmetric solution and (b)
Dm-symmetric trivial solutions (m divides n) presented in §5.2.
All these break bifurcations can be described by group-theoretic bifur-
cation theory (cf. Ikeda and Murota, 2002 [16]). The rule of bifurcation
depends on the integer number n. To be precise, it depends on the divisors
of the number n, and the bifurcation becomes increasingly hierarchical and
complex for n with more divisors.
A few examples are:
• If n is a prime number, then it can undergo the one and only course of
hierarchical bifurcations: Dn −→ D1 −→ C1.
• For n = 4 = 22, a hierarchy of subgroups expressing the rule of hi-
erarchical break bifurcations is presented in Fig. 13(a). As might be
readily apparent, in addition to the direct bifurcations in Fig. 5, sev-
eral secondary and tertiary bifurcations exist: D2-symmetric solution
branches into Dk,41 -symmetric ones (k = 1, . . . , 4) and Dk,4
1 -symmetric
one branches into C1-symmetric one.
• The hierarchy of subgroups for n = 6 = 2× 3 shown in Fig. 13(b) por-
trays a more complex hierarchy than that of n = 4 = 22 in Fig. 13(a).
These rules are sufficient in the description for break bifurcations of the
model, while the model in general undergoes more complex bifurcation due
to the presence of sustain bifurcation points as explained in Appendix D.2.
D.2 Sustain bifurcations
As mentioned in Appendix B.4, the sustain bifurcation point is a special
feature of the core–periphery model that leads to the extinction of city pop-
ulation of manufacturing labor. If we follow only feasible solutions, we must
switch into another feasible path. A sustain point may possibly appear in
48
Double
bifurcationSimple
bifurcation
D3,41
D2
D4 D1 D2,41
C1
D4,41
bifurcationSimple
bifurcationSimple
(a) Four cities (n = 4)
D3,6
1D1 D2,6
1 D4,6
1
1Ｄ1
1Ｄ1
1
Ｄ1
D5,6
1 D6,6
1
D3
D6
D3,6
2D2 D2,6
2C2
C1
(b) Six cities (n = 6)
Figure 13: Hierarchy of subgroups associated with hierarchical break bi-furcations (dark solid arrow, double bifurcation; thin solid arrow, simplebifurcation)
49
Double
Sustain point
bifurcationSimple
bifurcationSimple
bifurcationSimple
bifurcation
Sustainpoint
Sustainpoint
D3,41
D1
D2
D4
D2
D1
D1
D2,41
D4,41 C1
D4,41
Figure 14: Hierarchical, break and sustain bifurcations of the four cities(n = 4; solid arrow, break bifurcation; dashed arrow, sustain bifurcation)
any solutions other than the uniform population solution. The presence of
sustain bifurcation points must be meshed into the rule of break bifurcations
presented in Appendix D.1.
A possible course of hierarchical bifurcations is presented in Fig. 14, for
example, for the four cities (n = 4). From the trivial solution with uniform
population (λ = (1/4, 1/4, 1/4, 1/4)⊤) shown in the left, population distri-
bution patterns of various kinds are engendered via hierarchical bifurcations.
In this figure, we encounter sustain bifurcation points shown by the dashed
arrows, in addition to the double bifurcation shown by the dark solid arrow
and the simple bifurcations shown by the thin solid arrows. Cases other than
n = 4 can be treated similarly.
50
E Computational bifurcation theory
As we will see in §6, the solutions of the governing equation of the racetrack
economy involve several bifurcated paths that are quite complex. These
paths can be traced in a systematic and exhaustive manner by computational
bifurcation theory (cf. Crisfield, 1977 [5]). To be concrete, we employ the
following three numerical steps:
• Path tracing : In the path tracing of non-trivial solutions, we refer to
the incremental form of the equality equation (10), i.e.
F (u + δu, τ + δτ) − F (λ, τ)
= J(λ, τ)δλ +∂F
∂τ(λ, τ) δτ + h.o.t. = 0, (E.1)
where h.o.t. denotes higher order terms. At each equilibrium point
(λ, τ), another equilibrium point (λ+δλ, τ +δτ) can be found by solv-
ing26 (E.1) for (δλ, δτ) using a predictor–corrector (Newton–Raphson)
type method.
For the trivial solution(s), we need not carry out path tracing because
they are obtainable simply by symmetry consideration (cf. §5.2), al-
though singularity detection and branch switching must be conducted.
• Singularity detection: We carry out the eigenanalysis of the Jacobian
matrix J(λ, τ) at the solutions (λ, τ) to find the location of a critical
point, as a point at which one or more eigenvalues ei of J(λ, τ) become
zero.
For the uniform population solutions, the eigenanalysis can be con-
ducted using the explicit formulas in (22)∼(27). For the other solutions,
the numerical eigenanalysis of J , which is in general a non-symmetric
matrix, must be conducted.
• Branch switching : To obtain bifurcated paths branching from a bifur-
cation point, we use the so-called line search method.
26In the incremental equation (E.1), (λ, τ) is fixed, δλ is treated as an independentvariable, and δτ as a bifurcation parameter.
51
At a simple bifurcation point with a single zero eigenvalue, say e1=0
(M=1), a bifurcated path is to be sought in the direction of the crit-
ical eigenvector η1 associated with this zero eigenvalue. We employ
δλ = Cη1 as the initial value for the iteration to find a solution on a
bifurcated path, where the scaling constant C is to be specified perti-
nently in view of the convergence of the iteration.
At a double bifurcation point with two zero eigenvalues, say e1 = e2 = 0
(M=2), two (independent) bifurcated paths are to be sought in the
directions δλ = Cη(α1) and Cη(α2) in Proposition 2(iii).
We carry out an exhaustive search of a whole set of equilibrium paths of
the racetrack economy by repeating the three steps described above for sec-
ondary, tertiary, . . . bifurcated paths, until exhaustion of possible bifurcated
paths.
For the racetrack economy, it is possible to trace trivial solutions inde-
pendently, other than the uniform population solution, and to find bifurcated
paths from these trivial solutions (cf. §5.5).
52